ProtoQuant.com

Tangent Plane Instead of Fresnel Plane Wave

Fresnel made a huge imaginative effort to define the concept of plane wave, in order to use it for the accommodation of the principles and methods of classical mechanics. The suitability of application of these principles seemed to be out of question. The truth of the matter is that, what Fresnel used first was not mechanics, but the fact that in the case of light we have to deal with a periodical phenomenon. This property is all we need for the physical description of light in the manner of Fresnel. However, because the mathematics tells us that a periodical phenomenon is necessarily described by a second order differential equation, and as the second principle of dynamics has as expression a second order differential equation, it seems only natural that classical mechanics must be ruling even in the case of light. This is in essence the way in which classical mechanics got involved in the problem of light, and the main reason of rejection of the Newtonian image about light. However, if we think of a wave surface as of what it naturally is, i.e. a surface in space we don’t need classical mechanics in order to clarify the concept of light. All we need is a little differential geometry, and a clear thinking about what’s happening when the light meets matter. Indeed, the properties assigned to light have not been discovered but through the interaction between the light and matter. They could be obtained from a simple principle which can naturally give all the known properties of light. This principle is:

The wave surface is in permanent deformation

and replaces the Huygens’ Principle. Because of this we can consider it as a General Huygens Principle.

How does this principle work? Simply by the rules of the differential geometry: if a surface deforms we can locally recognize this phenomenon by the variation of its fundamental forms. The first fundamental form, i.e. the metric of the surface gives us the distance around a point of the surface as measured in its tangent plane in that point. The second fundamental form represents the variation of curvature in a point of the surface, again, when taking as reference the tangent plane to surface in that point (Struik, 1988). The physical properties of the surface can be entered by the six coefficients of the two fundamental forms of the surface and, obviously, by their variations. Therefore, if we take the wave surface as a surface from the point of view of the differential geometry, we don’t need the plane wave concept of Fresnel, because we have a natural plane at our disposal in each and every point of the wave surface: the tangent plane to the wave surface in that point.

Pages: 1 2 3 4 5 6